Marketing Research

Marketing ResearchAaker, Kumar, Day and LeoneTenth EditionInstructor’s Presentation Slides

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Marketing Research 10th Edition http://www.drvkumar.com/mr10/

Chapter Fourteen

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Correlation Analysis and Regression Analysis


Definitions

•Correlation analysis ▫Measures strength of the

relationship between two variables

•Correlation coefficient ▫Provides a measure of the

degree to which there is an association between two variables (X and Y)

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Regression Analysis

• Statistical technique that is used to relate two or more variables

• Objective is to build a regression model or a prediction equation relating the dependent variable to one or more independent variables

• The model can then be used to describe, predict, and control the variable of interest on the basis of the independent variables

• Multiple regression analysis - Regression analysis that involves more than one independent variable

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Correlation Analysis

• Pearson correlation coefficient

▫ Measures the degree to which there is a linear association between two interval-scaled variables

▫ A positive correlation reflects a tendency for a high value in one variable to be associated with a high value in the second

▫ A negative correlation reflects an association between a high value in one variable and a low value in the second variable

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Correlation Analysis (Contd.)

• Population correlation (p) - If the database includes an entire population

• Sample correlation (r) - If measure is based on a sample

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R lies between -1 < r < + 1

R = 0 ---> absence of linear association


Scatter Plots

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Scatter Plots (Contd.)

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Correlation Coefficient

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)(*)(),( YYXXyxCov ii

yS

YiY

xS

XiX

nxyr)(

**)1(

1

yx

xyxy SS

Covr

*

Simple Correlation Coefficient

Pearson Product-moment Correlation Coefficient


Determining Sample Correlation Coefficient

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Testing the Significance of the Correlation Coefficient

• Null hypothesis: Ho : p = 0

• Alternative hypothesis: Ha : p ≠ 0

• Test statistic

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96.170.01

2670.

2

tExample: n = 6 and r = .70

At = .05 , n-2 = 4 degrees of freedom, Critical value of t = 2.78Since 1.96<2.78, we fail to reject the null hypothesis.


Partial Correlation Coefficient

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Measure of association between two variables after controlling for the effects of one or more additional variables

)1(*)1(

*22,

YZXZ

YZXZXYZXY

rr

rrrr


Regression Analysis

Simple Linear Regression Model

Yi = βo + β1xi + εi Where

▫ Y = Dependent variable

▫ X =Independent variable

▫ β o = Model parameter that represents mean value of dependent variable (Y)

when the independent variable (X) is zero

▫ β1 = Model parameter that represents the slope that measures change in

mean value of dependent variable associated with a one-unit increase in the independent variable

▫εi = Error term that describes the effects on Yi of all factors other than

value of Xi

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Simple Linear Regression Model

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Simple Linear Regression Model – A Graphical Illustration

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Assumptions of the Simple Linear Regression Model• Error term is normally distributed (normality assumption)

• Mean of error term is zero [E(εi) = 0)

• Variance of error term is a constant and is independent of the values of X (constant variance assumption)

• Error terms are independent of each other (independent assumption)

• Values of the independent variable X are fixed (non-stochastic X)

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Estimating the Model Parameters• Calculate point estimate bo and b1 of unknown parameter βo and β1

• Obtain random sample and use this information from sample to estimate βo and β1

• Obtain a line of best "fit" for sample data points - least squares line

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Where

Predicted value of Yi ,


Residual Value

• bo and b1 minimize the residual or error sum of squares (SSE)

SSE = ei2 = ((yi - yi)2

= Σ [yi-(bo + b1xi)]2

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ei = yi - yi

= yi - (bo + b1 xi)

• Difference between the actual and predicted values

• Estimate of the error in the population


Standard Error

• Mean Square Error

• Standard Error of b1

• Standard Error of b0

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Testing the Significance of Independent Variables• Null Hypothesis

▫ There is no linear relationship between the independent & dependent variables

• Alternative Hypothesis

▫ There is a linear relationship between the independent & dependent variables

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Ha: β1 ≠ 0

H0: β1 = 0


Testing the Significance of Independent Variables (Contd.)

• Test Statistic t = b1 - β1

sb1

• Degrees of Freedom V = n – 2

• Testing for a Type II Error

Ho: β1 = 0

Ha: β1 ≠ 0

• Decision Rule

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Reject ho: β1 = 0 if α > p value


Sum of Squares

SST Sum of squared prediction error that would beobtained if we do not use x to predict y

SSE Sum of squared prediction error that is obtained when we use x to predict y

SSM Reduction in sum of squared prediction error that has been accomplished using x in predicting y

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Predicting the Dependent Variable

• Dependent variable, yi = bo + bixi • Error of prediction is yi – y

• Total variation (SST)= Explained variation (SSM) + Unexplained variation (SSE)

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(Yi - Y)2 = (Yi - Y)2 + (Yi – Yi)2

Coefficient of Determination (r2)• Measure of regression model's ability to predict

r2 = (SST - SSE) / SST= SSM / SST= Explained Variation / Total Variation


Multiple Regression

• A linear combination of predictor factors is used to predict the outcome or response factors

• The general form of the multiple regression model is explained as:

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where β1 , β2, . . . , βk are regression coefficients associated with the independent variables X1, X2, . . . , Xk and

ε is the error or residual.


Multiple Regression (Contd.)

•The prediction equation in multiple regression analysis is

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Ŷ = α + b1X1 + b2X2 + …….+bkXk

where Ŷ is the predicted Y score and b1 . . . , bk are the partial regression coefficients.


Partial Regression Coefficients

• b 1 is the expected change in Y when X1 is

changed by one unit, keeping X 2 constant or

controlling for its effects.

• b 2 is the expected change in Y for a unit

change in X2, when X1 is held constant.

• If X1 and X2 are each changed by one unit, the

expected change in Y will be (b1 / b2)

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Y = α + b1X1 + b2X2 + error


Evaluating the Importance of Independent Variables

• Consider t-value for βi's

• Use beta coefficients when independent variables are in different units of measurement

Standardized βi = bi Standard deviation of

xi

Standard deviation of Y

• Check for multicollinearity

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Stepwise Regression• Predictor variables enter or are removed from

the regression equation one at a time

• Forward Addition▫Start with no predictor variables in regression

equation

i.e. y = βo + ε

▫Add variables if they meet certain criteria in terms of F-ratio

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Stepwise Regression (Contd.)

• Backward Elimination

▫Start with full regression equation

i.e. y = βo + β1x1 + β2 x2 ...+ βr xr + ε

▫Remove predictors based on F- ratio

• Stepwise Method

▫Forward addition method is combined with removal of predictors that no longer meet specified criteria at each step

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Residual Plots

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Random distribution of residuals

Nonlinear pattern of residuals

HeteroskedasticityAutocorrelation


Predictive Validity• Examines whether any model estimated with one set of data continues to

hold good on comparable data not used in the estimation.

• Estimation Methods

1. The data are split into the estimation sample (with more than half of the total

sample) and the validation sample, and the coefficients from the two samples

are compared.

2. The coefficients from the estimated model are applied to the data in the

validation sample to predict the values of the dependent variable Yi in the

validation sample, and then the model fit is assessed.

3. The sample is split into halves – estimation sample and validation sample for

conducting cross-validation. The roles of the estimation and validation halves

are then reversed, and the cross-validation is repeated

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Regression with Dummy Variables

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Yi = a + b1D1 + b2D2 + b3D3 + error

• For rational buyer, Ŷi = a

• For brand-loyal consumers, Ŷi = a + b1

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